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SELF-DIFFUSIOPHORESIS AND BIOLOGICAL MOTILITY
Informal Seminar, University of Oxford, 8th June 2011
Department of Physics & AstronomyUniversity of Pennsylvania
ACTIN BASED PROPULSION
Listeria monocytogenes
Courtesy of Julie Theriot http://cmgm.stanford.edu/theriot/movies.htm
ACTIN AND LISTERIA MOTILITYLife cycle of
Listeria monocytogenes
sphere
IN-VITRO REALISATIONS
Courtesy of Julie Theriot http://cmgm.stanford.edu/
theriot/movies.htm
“All” you need isActin and buffer w/ATP
Arp2/3 makes new growing endsCapping protein kills them off
ADF/cofilin severs filaments
Profilin converts ADP-G-actin to ATP-G-actinBead coated with ActA, VCA, activates Arp2/3
how does self-assembly of actin into a branched structure lead to motility?
sphere
IN-VITRO REALISATIONS
Courtesy of Julie Theriot http://cmgm.stanford.edu/
theriot/movies.htm
“All” you need isActin and buffer w/ATP
Arp2/3 makes new growing endsCapping protein kills them off
ADF/cofilin severs filaments
Profilin converts ADP-G-actin to ATP-G-actinBead coated with ActA, VCA, activates Arp2/3
how does self-assembly of actin into a branched structure lead to motility?
BROWNIAN DYNAMICS SIMULATIONS
Polymerisation at + end ( )
Depolymerisation at - end ( )
Branching ( )
Debranching ( )
Capping
BROWNIAN DYNAMICS SIMULATIONS
2D projection of 3D simulation
Motion allowed in direction only
BROWNIAN DYNAMICS SIMULATIONS
2D projection of 3D simulation
Motion allowed in direction only
disk
Courtesy of J. Theriot
ACTIN CONCENTRATION GRADIENT
Disk activates Arp2/3, which recruits F-actin
Concentration of F-actin is high behind the disk compared to average
If the disk repels actin then it will move forwards to avoid F-actin
In real systems the concentration gradient is even bigger; mechanism should still apply
ASYMMETRIC CELL DIVISION
Most cells divide symmetrically
What is the mechanism for chromosomal motility?
Caulobacter crescentus and Vibrio cholerae
divide asymmetrically
DISASSEMBLY DRIVEN MOTILITY
Caulobacter crescentus
Courtesy of C. W. Shebelut, J. M. Guberman and Z. Gitai
How does the chromosome move across the cell during chromosomal segregation in certain asymmetric bacteria?
DISASSEMBLY DRIVEN MOTILITY
Caulobacter crescentus
Courtesy of C. W. Shebelut, J. M. Guberman and Z. Gitai
How does the chromosome move across the cell during chromosomal segregation in certain asymmetric bacteria?
CHROMOSOMAL SEGREGATION IN C. CRESCENTUS AND V. CHOLERAE
Courtesy of C. W. Shebelut, J. M. Guberman and Z. Gitai
Caulobacter crescentus
We are interested in the 4th stage
How does the chromosome (ori) scoot across the cell?
Courtesy of Popular Logistics
VIBRIO CHOLERAE
ReplicationParB on origin
attaches to ParA
ParA disassembles and origin moves
Origin and terminus switch places
A CLOSER LOOK AT THE PROCESS
Origin is decorated with ParB which binds to and hydrolyses ParA
ParA filament structure depolymerises and drags ParB along
origin 1
origin 2
origin 1
ori2/ParBterminus
ParA
CONCENTRATION GRADIENT DRIVES MOTION
System uses depolymerisation to create a steady-state concentration gradient to move up
Courtesy of J. Theriot
BIOLOGICAL MOTILITY
Two examples of motion involving the assembly or disassembly of filaments
Simulations replicate interaction with filaments - suggests motion in a filament concentration gradient
but without fluid flow
PARTICLE MOTION IN A CONCENTRATION GRADIENT
Particle interacts with the concentration field and moves
PARTICLE MOTION IN A CONCENTRATION GRADIENT
Particle interacts with the concentration field and moves down the gradient if it is repelled
PARTICLE MOTION IN A CONCENTRATION GRADIENT
Particle interacts with the concentration field and moves up the gradient if it is attracted
PARTICLE MOTION IN A CONCENTRATION GRADIENT
Particle interacts with the concentration field and moves up the gradient if it is attracted
A PARED DOWN MODEL
Spherical particle
Axisymmetry, steady state
Single solute species
Purely radial potential with compact support
Zero Reynolds number
PARTICLE MOTION IN A CONCENTRATION GRADIENT
Motion involves a balance between diffusion and advection
Diffusion dominated Advection dominated
BOUNDARY LAYER ANALYSIS
The interaction occurs close to the surfacein a boundary layer of thickness
Solve Stokes in the upper half space
BOUNDARY LAYER ANALYSIS
The interaction occurs close to the surfacein a boundary layer of thickness
Solve Stokes in the upper half space
Tangential slip velocity
diffusiophoretic mobility (Derjaguin)
SQUIRMERS
Slip velocity provides an inner boundary condition for the exterior flow
General solution provided by Lighthill’s squirmer model
Matching the boundary condition gives the speed
first Legendre coefficient
SOLUTE PROFILE
Outside the boundary layer the solute is conserved
Rate of transport across the boundary layer equals rate of production
SOLUTE PROFILE
Outside the boundary layer the solute is conserved
Rate of transport across the boundary layer equals rate of production
solve pointwise
DIGRESSION
Boundary layer analysis neglects
boundary layer
not constant
fluid continuity; radial slip
DIGRESSION
Boundary layer analysis neglects
boundary layer
not constant
fluid continuity; radial slip
Scaling
generically small
DIGRESSION
Boundary layer analysis neglects
fluid continuity; radial slip
topology of the sphere
boundary layer
DIGRESSION
Boundary layer analysis neglects
fluid continuity; radial slip
topology of the sphere
Neglecting the radial slip is not a good approximation near
consequence of topology rather than axisymmetry (Poincaré-Hopf)
DIGRESSION
Boundary layer analysis neglects
Such a simple problem can be solved exactly
Poincaré-Hopf
boundary layer
fluid continuity; radial slip
topology of the sphere
Spherical particle
Axisymmetry, steady state
Single solute species
Purely radial potential
Zero Reynolds number
Zero Péclet number
DIGRESSION
Boundary layer analysis neglects
Such a simple problem can be solved exactly
Poincaré-Hopf
boundary layer
fluid continuity; radial slip
topology of the sphere
correction
E.g., the speed is
DIGRESSION
Boundary layer analysis neglects
Such a simple problem can be solved exactly
Poincaré-Hopf
boundary layer
fluid continuity; radial slip
topology of the sphere
scaling
E.g., the speed is
BOUNDARY LAYER FLOW
1.02 1.04 1.06 1.08 1.10
�0.8
�0.6
�0.4
�0.2
1.02 1.04 1.06 1.08 1.10
0.05
0.10
0.15
0.20
BOUNDARY LAYER FLOW
1.5 2.0 2.5 3.0 3.5
�1.0
�0.8
�0.6
�0.4
�0.2
0.2
0.4
1.5 2.0 2.5 3.0 3.5
0.2
0.4
0.6
0.8
WHAT CHANGES IF THE PÉCLET NUMBER IS LARGE?
concentration gradients drive tangential flow in a thin boundary layer
Basic mechanism remains unchanged
postulate
WHAT CHANGES IF THE PÉCLET NUMBER IS LARGE?
concentration gradients drive tangential flow in a thin boundary layer
Basic mechanism remains unchanged
if the solute does not diffuse then it will only be found where it is producedtangential slip only generated within the active patchradial influx at the boundary and outflux from the interior
But ...
postulate
SOLUTE TRANSPORT
Outside the boundary layer the solute is conserved
Rate of transport across the boundary layer equals rate of production
SOLUTE TRANSPORT
Outside the boundary layer the solute is conserved
Rate of transport across the boundary layer equals rate of production
radial slip is important
RADIAL SLIP
Radial outflux
Tangential influx
conserved away from
SO, WHAT’S DIFFERENT?
SO, WHAT’S DIFFERENT?
SO, WHAT’S DIFFERENT?
Transport balance
SO, WHAT’S DIFFERENT?
Transport balance
by fluid continuityby postulate
SO, WHAT’S DIFFERENT?
SO, WHAT’S DIFFERENT?
first Legendre coefficient of activity
tangential flux is conserved
by, e.g., Stone & Samuel
SO, WHAT’S DIFFERENT?
does not vanish for total coverage
SO, WHAT’S DIFFERENT?
does not vanish for total coverage
is state of total coverage unstable?if so, what is the critical Péclet number for the instability?
DIFFERENT SCENARIOS
repulsive producer
attractive consumerrepulsive consumer
attractive producer
DIFFERENT SCENARIOS
repulsive producer
attractive consumerrepulsive consumer
attractive producer
FLUID DROPLETS
FLUID DROPLETS
Droplets containing a catalyst dispersed in a bulk fuel
Fuel hydrolysed at the surface
Waste product accumulates on the surface and is released at the rear
Self-maintained surface tension gradients drive Marangoni flows
FLUID DROPLETS
Spherical particle
Axisymmetry, steady state
Single solute species
Purely radial potential with compact support
Zero Reynolds number
Zero Péclet number
FLUID DROPLETS
Spherical particle
Axisymmetry, steady state
Single solute species
Purely radial potential with compact support
Zero Reynolds number
Zero Péclet number
Stress balance at the interface
Marangoni stress
surface tension
mean curvatureexternal fluid stress
internal fluid stress
surface normal
FLUID DROPLETS
Solve as before; e.g., the speed is
Spherical particle
Axisymmetry, steady state
Single solute species
Purely radial potential with compact support
Zero Reynolds number
Zero Péclet number
FLUID DROPLETS
Solve as before; e.g., the speed is
Spherical particle
Axisymmetry, steady state
Single solute species
Purely radial potential with compact support
Zero Reynolds number
Zero Péclet number
independent of particle size
limit
(solid particle)
limit
(gas bubble)
proportional to particle radius
ACTIVITY ON THE INSIDE
No need for a favourable environment -- take everything you need with you!
“Clean” system; everything is internal
Only interaction between droplets is hydrodynamic
ACTIVITY ON THE INSIDE
Scaling
same as the “gas bubble”
No need for a favourable environment -- take everything you need with you!
“Clean” system; everything is internal
Only interaction between droplets is hydrodynamic
Solve as before; e.g., the speed is
SURFACE TENSION GRADIENTS
Stress balance at the interface
Marangoni stress
surface tension
mean curvatureexternal fluid stress
internal fluid stress
surface normal
Activity due to a surface active catalyst
Surface adsorbed species lower the surface tension
Chemical reaction near surface produces local heating; lowers surface tension
Non-uniform surface tension drives Marangoni flows
SURFACE TENSION GRADIENTS
Activity due to a surface active catalyst
Surface adsorbed species lower the surface tension
Chemical reaction near surface produces local heating; lowers surface tension
Non-uniform surface tension drives Marangoni flows
Additional contribution to the speed
typically surface tension is lowered so that is negativeMarangoni flows then oppose
self-diffusiophoresis
SURFACE TENSION GRADIENTS
Activity due to a surface active catalyst
Surface adsorbed species lower the surface tension
Chemical reaction near surface produces local heating; lowers surface tension
Non-uniform surface tension drives Marangoni flows
ratio
can this be made small?
Additional contribution to the speed
Stress balance at the interface
Marangoni stress
surface tension
mean curvatureexternal fluid stress
internal fluid stress
surface normal
Particle is a fluid droplet -- no reason why it won’t deform
Normal stress balance is really an equation for the drop shape
DROPLET DEFORMATION
droplet remains approximately spherical provided
THANKS!
FUNDING
We are grateful to Ed Banigan and Kun-Chun Lee for beneficial discussions and to Randy Kamien for his support and encouragement
Fluid drops can move due to internal motor
“Clean” system
Faster than a solid particle
High Péclet number relevant to biological motility
Different scaling with activity, dependence on coverage and successful strategies
Recommended